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arXiv · 2402.15545

Global weak solutions of a Hamiltonian regularised Burgers equation

Billel Guelmame·Stéphane Junca·Didier Clamond·Robert L. Pego
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Abstract

A nondispersive, conservative regularisation of the inviscid Burgers equation is proposed and studied. Inspired by a related regularisation of the shallow water system recently introduced by Clamond and Dutykh, the new regularisation provides a family of Galilean-invariant interpolants between the inviscid Burgers equation and the Hunter-Saxton equation. It admits weakly singular regularised shocks and cusped traveling-wave weak solutions. The breakdown of local smooth solutions is demonstrated, and the existence of two types of global weak solutions, conserving or dissipating an $H^1$ energy, is established. Dissipative solutions satisfy an Oleinik inequality like entropy solutions of the inviscid Burgers equation. As the regularisation scale parameter $\ell$ tends to $0$ or $\infty$, limits of dissipative solutions are shown to satisfy the inviscid Burgers or Hunter-Saxton equation respectively, forced by an unknown remaining term.

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Billel Guelmame, Stéphane Junca, Didier Clamond, Robert L. Pego. 2024-02-23. Global weak solutions of a Hamiltonian regularised Burgers equation. https://doi.org/10.1007/s10884-022-10171-0

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